Properties

Label 1323.25
Modulus $1323$
Conductor $1323$
Order $63$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1323, base_ring=CyclotomicField(126))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([70,48]))
 
pari: [g,chi] = znchar(Mod(25,1323))
 

Basic properties

Modulus: \(1323\)
Conductor: \(1323\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(63\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1323.ca

\(\chi_{1323}(25,\cdot)\) \(\chi_{1323}(58,\cdot)\) \(\chi_{1323}(88,\cdot)\) \(\chi_{1323}(121,\cdot)\) \(\chi_{1323}(151,\cdot)\) \(\chi_{1323}(184,\cdot)\) \(\chi_{1323}(247,\cdot)\) \(\chi_{1323}(277,\cdot)\) \(\chi_{1323}(310,\cdot)\) \(\chi_{1323}(340,\cdot)\) \(\chi_{1323}(403,\cdot)\) \(\chi_{1323}(436,\cdot)\) \(\chi_{1323}(466,\cdot)\) \(\chi_{1323}(499,\cdot)\) \(\chi_{1323}(529,\cdot)\) \(\chi_{1323}(562,\cdot)\) \(\chi_{1323}(592,\cdot)\) \(\chi_{1323}(625,\cdot)\) \(\chi_{1323}(688,\cdot)\) \(\chi_{1323}(718,\cdot)\) \(\chi_{1323}(751,\cdot)\) \(\chi_{1323}(781,\cdot)\) \(\chi_{1323}(844,\cdot)\) \(\chi_{1323}(877,\cdot)\) \(\chi_{1323}(907,\cdot)\) \(\chi_{1323}(940,\cdot)\) \(\chi_{1323}(970,\cdot)\) \(\chi_{1323}(1003,\cdot)\) \(\chi_{1323}(1033,\cdot)\) \(\chi_{1323}(1066,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((785,1081)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{8}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{29}{63}\right)\)\(e\left(\frac{58}{63}\right)\)\(e\left(\frac{52}{63}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{29}{63}\right)\)\(e\left(\frac{1}{63}\right)\)\(e\left(\frac{53}{63}\right)\)\(e\left(\frac{6}{7}\right)\)\(1\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 63 polynomial